What is Cluster Analysis? COMP 465: Data Mining Clustering Basics. Applications of Cluster Analysis. Clustering: Application Examples 3/17/2015
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1 // What is Cluster Analysis? COMP : Data Mining Clustering Basics Slides Adapted From : Jiawei Han, Micheline Kamber & Jian Pei Data Mining: Concepts and Techniques, rd ed. Cluster: A collection of data objects similar (or related) to one another within the same group dissimilar (or unrelated) to the objects in other groups Cluster analysis (or clustering, data segmentation, ) Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters Unsupervised learning: no predefined classes (i.e., learning by observations vs. learning by examples: supervised) Typical applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms Applications of Cluster Analysis Data reduction Summarization: Preprocessing for regression, PCA, classification, and association analysis Compression: Image processing: vector quantization Hypothesis generation and testing Prediction based on groups Cluster & find characteristics/patterns for each group Finding K-nearest eighbors Localizing search to one or a small number of clusters Outlier detection: Outliers are often viewed as those far away from any cluster Clustering: Application Examples Biology: taxonomy of living things: kingdom, phylum, class, order, family, genus and species Information retrieval: document clustering Land use: Identification of areas of similar land use in an earth observation database Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults Climate: understanding earth climate, find patterns of atmospheric and ocean Economic Science: market resarch
2 // Basic Steps to Develop a Clustering Task Feature selection Select info concerning the task of interest Minimal information redundancy Proximity measure Similarity of two feature vectors Clustering criterion Expressed via a cost function or some rules Clustering algorithms Choice of algorithms Validation of the results Validation test (also, clustering tendency test) Interpretation of the results Integration with applications Quality: What Is Good Clustering? A good clustering method will produce high quality clusters high intra-class similarity: cohesive within clusters low inter-class similarity: distinctive between clusters The quality of a clustering method depends on the similarity measure used by the method its implementation, and Its ability to discover some or all of the hidden patterns Measure the Quality of Clustering Dissimilarity/Similarity metric Similarity is expressed in terms of a distance function, typically metric: d(i, j) The definitions of distance functions are usually rather different for interval-scaled, boolean, categorical, ordinal ratio, and vector variables Weights should be associated with different variables based on applications and data semantics Quality of clustering: There is usually a separate quality function that measures the goodness of a cluster. It is hard to define similar enough or good enough The answer is typically highly subjective Considerations for Cluster Analysis Partitioning criteria Single level vs. hierarchical partitioning (often, multi-level hierarchical partitioning is desirable) Separation of clusters Exclusive (e.g., one customer belongs to only one region) vs. non-exclusive (e.g., one document may belong to more than one class) Similarity measure Distance-based (e.g., Euclidian, road network, vector) vs. connectivity-based (e.g., density or contiguity) Clustering space Full space (often when low dimensional) vs. subspaces (often in highdimensional clustering)
3 // Requirements and Challenges Scalability Clustering all the data instead of only on samples Ability to deal with different types of attributes umerical, binary, categorical, ordinal, linked, and mixture of these Constraint-based clustering User may give inputs on constraints Use domain knowledge to determine input parameters Interpretability and usability Others Discovery of clusters with arbitrary shape Ability to deal with noisy data Incremental clustering and insensitivity to input order High dimensionality Major Clustering Approaches (I) Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errors Typical methods: k-means, k-medoids, CLARAS Hierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterion Typical methods: Diana, Agnes, BIRCH, CAMELEO Density-based approach: Based on connectivity and density functions Typical methods: DBSAC, OPTICS, DenClue Grid-based approach: based on a multiple-level granularity structure Typical methods: STIG, WaveCluster, CLIQUE Partitioning Algorithms: Basic Concept The K-Means Clustering Method Partitioning method: Partitioning a database D of n objects into a set of k clusters, such that the sum of squared distances is minimized (where c i is the centroid or medoid of cluster C i ) k E i p C ( d( p, c )) i i Given k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen, Lloyd / ): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw ): Each cluster is represented by one of the objects in the cluster Given k, the k-means algorithm is implemented in four steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the clusters of the current partitioning (the centroid is the center, i.e., mean point, of the cluster) Assign each object to the cluster with the nearest seed point Go back to Step, stop when the assignment does not change
4 // An Example of K-Means Clustering Comments on the K-Means Method K= Strength: Efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. ormally, k, t << n. The initial data set Until no change Arbitrarily partition objects into k groups Partition objects into k nonempty subsets Repeat Compute centroid (i.e., mean point) for each partition Assign each object to the cluster of its nearest centroid Loop if needed Update the cluster centroids Update the cluster centroids Reassign objects Comparing: PAM: O(k(n-k) ), CLARA: O(ks + k(n-k)) Comment: Often terminates at a local optimal Weakness Applicable only to objects in a continuous n-dimensional space Using the k-modes method for categorical data In comparison, k-medoids can be applied to a wide range of data eed to specify k, the number of clusters, in advance (there are ways to automatically determine the best k (see Hastie et al., ) Sensitive to noisy data and outliers ot suitable to discover clusters with non-convex shapes Variations of the K-Means Method What Is the Problem of the K-Means Method? Most of the variants of the k-means differ in Selection of the initial k means Dissimilarity calculations Strategies to calculate cluster means Handling categorical data: k-modes Replacing means of clusters with modes Using new dissimilarity measures to deal with categorical objects Using a frequency-based method to update modes of clusters A mixture of categorical and numerical data: k-prototype method The k-means algorithm is sensitive to outliers! Since an object with an extremely large value may substantially distort the distribution of the data K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster
5 // PAM: A Typical K-Medoids Algorithm The K-Medoid Clustering Method K= Do loop Until no change Arbitrary choose k object as initial medoids Swapping O and O ramdom If quality is improved. Total Cost = Assign each remainin g object to nearest medoids Compute total cost of swapping Total Cost = Randomly select a nonmedoid object,o ramdom K-Medoids Clustering: Find representative objects (medoids) in clusters PAM (Partitioning Around Medoids, Kaufmann & Rousseeuw ) Starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering PAM works effectively for small data sets, but does not scale well for large data sets (due to the computational complexity) Efficiency improvement on PAM CLARA (Kaufmann & Rousseeuw, ): PAM on samples CLARAS (g & Han, ): Randomized re-sampling Hierarchical Clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step Step Step Step Step a b c d e a b d e c d e a b c d e Step Step Step Step Step agglomerative (AGES) divisive (DIAA) AGES (Agglomerative esting) Introduced in Kaufmann and Rousseeuw () Implemented in statistical packages, e.g., Splus Use the single-link method and the dissimilarity matrix Merge nodes that have the least dissimilarity Go on in a non-descending fashion Eventually all nodes belong to the same cluster
6 // Dendrogram: Shows How Clusters are Merged DIAA (Divisive Analysis) Decompose data objects into several levels of nested partitioning (tree of clusters), called a dendrogram A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster Introduced in Kaufmann and Rousseeuw () Implemented in statistical analysis packages, e.g., Splus Inverse order of AGES Eventually each node forms a cluster on its own Distance between Clusters X X Centroid, Radius and Diameter of a Cluster (for numerical data sets) X Single link: smallest distance between an element in one cluster and an element in the other, i.e., dist(k i, K j ) = min(t ip, t jq ) Complete link: largest distance between an element in one cluster and an element in the other, i.e., dist(k i, K j ) = max(t ip, t jq ) Average: avg distance between an element in one cluster and an element in the other, i.e., dist(k i, K j ) = avg(t ip, t jq ) Centroid: distance between the centroids of two clusters, i.e., dist(k i, K j ) = dist(c i, C j ) Medoid: distance between the medoids of two clusters, i.e., dist(k i, K j ) = dist(m i, M j ) Medoid: a chosen, centrally located object in the cluster Centroid: the middle of a cluster C m ( tip) i Radius: square root of average distance from any point of the cluster to its centroid ( t c m ) R i ip m Diameter: square root of average mean squared distance between all pairs of points in the cluster ( t t ) D i i ip iq m ( )
7 // Extensions to Hierarchical Clustering Major weakness of agglomerative clustering methods Can never undo what was done previously Do not scale well: time complexity of at least O(n ), where n is the number of total objects Integration of hierarchical & distance-based clustering BIRCH (): uses CF-tree and incrementally adjusts the quality of sub-clusters CHAMELEO (): hierarchical clustering using dynamic modeling BIRCH (Balanced Iterative Reducing and Clustering Using Hierarchies) Zhang, Ramakrishnan & Livny, SIGMOD Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering Phase : scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) Phase : use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans Weakness: handles only numeric data, and sensitive to the order of the data record Clustering Feature Vector in BIRCH CF-Tree in BIRCH Clustering Feature (CF): CF = (, LS, SS) : umber of data points LS: linear sum of points: X i i SS: square sum of points X i i CF = (, (,),(,)) (,) (,) (,) (,) (,) Clustering feature: Summary of the statistics for a given subcluster: the -th, st, and nd moments of the subcluster from the statistical point of view Registers crucial measurements for computing cluster and utilizes storage efficiently A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering A non-leaf node in a tree has descendants or children The non-leaf nodes store sums of the CFs of their children A CF tree has two parameters Branching factor: max # of children Threshold: max diameter of sub-clusters stored at the leaf nodes
8 // The CF Tree Structure Root The Birch Algorithm B = L = CF child CF child CF child CF child Cluster Diameter ( x x ) n( n ) i j CF child CF child on-leaf node CF child Leaf node CF child Leaf node prev CF CF CF next prev CF CF CF next For each point in the input Find closest leaf entry Add point to leaf entry and update CF If entry diameter > max_diameter, then split leaf, and possibly parents Algorithm is O(n) Concerns Sensitive to insertion order of data points Since we fix the size of leaf nodes, some clusters may not be so natural Clusters tend to be spherical given the radius and diameter measures CHAMELEO: Hierarchical Clustering Using Dynamic Modeling () CHAMELEO: G. Karypis, E. H. Han, and V. Kumar, Measures the similarity based on a dynamic model Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters Graph-based, and a two-phase algorithm. Use a graph-partitioning algorithm: cluster objects into a large number of relatively small sub-clusters. Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these subclusters K Graphs & Interconnectivity k-nearest graphs from an original data in D: EC {Ci,Cj } : The absolute inter-connectivity between C i and C j : the sum of the weight of the edges that connect vertices in C i to vertices in C j Internal inter-connectivity of a cluster C i : the size of its min-cut bisector EC Ci (i.e., the weighted sum of edges that partition the graph into two roughly equal parts) Relative Inter-connectivity (RI):
9 // Relative Closeness & Merge of Sub-Clusters Overall Framework of CHAMELEO Relative closeness between a pair of clusters C i and C j : the absolute closeness between C i and C j normalized w.r.t. the internal closeness of the two clusters C i and C j Construct (K-) Sparse Graph Partition the Graph and are the average weights of the edges that belong in the min-cut bisector of clusters C i and C j, respectively, and is the average weight of the edges that connect vertices in C i to vertices in C j Merge Sub-Clusters: Merges only those pairs of clusters whose RI and RC are both above some user-specified thresholds Merge those maximizing the function that combines RI and RC Data Set K- Graph P and q are connected if q is among the top k closest neighbors of p Final Clusters Merge Partition Relative interconnectivity: connectivity of c and c over internal connectivity Relative closeness: closeness of c and c over internal closeness CHAMELEO (Clustering Complex Objects) Probabilistic Hierarchical Clustering Algorithmic hierarchical clustering ontrivial to choose a good distance measure Hard to handle missing attribute values Optimization goal not clear: heuristic, local search Probabilistic hierarchical clustering Use probabilistic models to measure distances between clusters Generative model: Regard the set of data objects to be clustered as a sample of the underlying data generation mechanism to be analyzed Easy to understand, same efficiency as algorithmic agglomerative clustering method, can handle partially observed data In practice, assume the generative models adopt common distribution functions, e.g., Gaussian distribution or Bernoulli distribution, governed by parameters
10 // Generative Model Gaussian Distribution Given a set of -D points X = {x,, x n } for clustering analysis & assuming they are generated by a Gaussian distribution: The probability that a point x i X is generated by the model Bean machine: drop ball with pins The likelihood that X is generated by the model: The task of learning the generative model: find the parameters μ and σ such that the maximum likelihood -d Gaussian From wikipedia and -d Gaussian A Probabilistic Hierarchical Clustering Algorithm For a set of objects partitioned into m clusters C,...,C m, the quality can be measured by, where P() is the maximum likelihood If we merge two clusters C j and C j into a cluster C j C j, then, the change in quality of the overall clustering is Distance between clusters C and C : If dist(c i, C j ) <, merge C i and C j Summary Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be computed for various types of data Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, gridbased methods, and model-based methods K-means and K-medoids algorithms are popular partitioningbased clustering algorithms Birch and Chameleon are interesting hierarchical clustering algorithms, and there are also probabilistic hierarchical clustering algorithms
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